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What is the area under the curve y = x^4 from x = 0 to x = 1?

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Question: What is the area under the curve y = x^4 from x = 0 to x = 1?

Options:

  1. 1/5
  2. 1/4
  3. 1/3
  4. 1/2

Correct Answer: 1/5

Solution: 

The area under the curve y = x^4 from x = 0 to x = 1 is given by ∫(from 0 to 1) x^4 dx = [x^5/5] from 0 to 1 = 1/5.

What is the area under the curve y = x^4 from x = 0 to x = 1?

Practice Questions

Q1
What is the area under the curve y = x^4 from x = 0 to x = 1?
  1. 1/5
  2. 1/4
  3. 1/3
  4. 1/2

Questions & Step-by-Step Solutions

What is the area under the curve y = x^4 from x = 0 to x = 1?
Correct Answer: 1/5
  • Step 1: Understand that we want to find the area under the curve of the function y = x^4 between x = 0 and x = 1.
  • Step 2: To find the area, we need to use integration. We will set up the integral as ∫(from 0 to 1) x^4 dx.
  • Step 3: Calculate the integral of x^4. The integral of x^n is (x^(n+1))/(n+1). Here, n = 4, so we get (x^(4+1))/(4+1) = x^5/5.
  • Step 4: Now we need to evaluate this integral from 0 to 1. This means we will calculate [x^5/5] from 0 to 1.
  • Step 5: First, plug in the upper limit (1): (1^5)/5 = 1/5.
  • Step 6: Next, plug in the lower limit (0): (0^5)/5 = 0.
  • Step 7: Now, subtract the lower limit result from the upper limit result: (1/5) - (0) = 1/5.
  • Step 8: Therefore, the area under the curve y = x^4 from x = 0 to x = 1 is 1/5.
  • Definite Integral – The question tests the understanding of calculating the area under a curve using definite integrals.
  • Polynomial Functions – The function y = x^4 is a polynomial, and the question assesses the ability to integrate polynomial functions.
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